Question

Solve the inequality: \(3 x+4<8-x\)

Short answer

x < 1

Step-by-step solution

Isolate Variables

Start by moving all the terms with x to one side of the inequality. Add x to both sides: \[3x + 4 + x < 8\]

Combine Like Terms

Combine the x terms on the left side: \[4x + 4 < 8\]

Isolate x

Subtract 4 from both sides to get the terms with x alone: \[4x < 4\]

Solve for x

Divide both sides by 4 to solve for x: \[x < 1\]

Key concepts

Isolate Variables

One of the first steps in solving inequalities is to isolate the variable of interest, which in our case is **x**. By isolating variables, we mean getting all the terms with **x** on one side of the inequality. This makes it easier to manipulate and solve the inequality.

**Example Procedure:**

• Start with the inequality: \[3x + 4 < 8 - x\]
• Move the **x** terms to one side by adding **x** to both sides: \[3x + 4 + x < 8\]

Now we have simplified the inequality to a single side containing all the **x** terms.
This setup is crucial as it allows us to simplify and eventually solve for **x** efficiently.

Combine Like Terms

After isolating the variable on one side, the next step is to combine like terms. Like terms are terms that involve the same variable raised to the same power.

For our example, we have combined like terms as follows: \[3x + 4 + x < 8\] Combining the **x** terms, we get: \[4x + 4 < 8\] This step is essential for simplifying the inequality, making it easier to handle and solve. When combining like terms, make sure to:

• Add or subtract the coefficients of the variables.
• Ensure like terms are on the same side of the inequality.

Smoothly combining like terms paves the way for the next step, further isolating the variable.” }, {